What is a linear sequence?

A linear sequence is a sequence that has a common difference. This means the sequence increases or decreases by the same amount each time. For example 4, 6, 8, 10... and 15, 11, 7, 3... are linear sequences but 1,4,9,16... is not.

What is an nth term?

You will by now be familiar with continuing a sequence by finding the difference, this is called a term to term rule. An nth term is a position-to-term rule, it lets you find the value in the sequence if you know it's position.

For example, if you had an nth term of 4n - 5, you could find the 10^{th} term by setting n=10 and substituting in to the nth term to get 4(10)-5 = 35.

The 10th term in the sequence would be 35.

What do linear nth terms look like?

All linear nth terms have the same structure, an + b, where a and b are numbers to be found.

**Example 1: Find the nth term for 5, 7, 9, 11...**

**Step 1**: Find the common difference. (a)

We can see that this sequence goes up by 2 each time. This means our nth term will start with 2n

**Step 2**: Find the value of b.

We can use a method here called the zero term. Just ask 'What number would come before the first term?'. In this example, the number would be 3 (the numbers change by 2 each time)

Our Nth term will be 2n + 3

**Example 2: Find the nth term for -1, 3, 7, 11...**

**Step 1**: Find the common difference. (a)

We can see that this sequence goes up by 4 each time. This means our nth term will start with 4n

**Step 2**: Find the value of b.

'What number would come before the first term?'. In this example, the number would be -5 (the numbers change by 4 each time)

Our Nth term will be 4n - 5

**Example 3: Find the nth term for 7, 4, 1, -2...**

**Step 1**: Find the common difference. (a)

We can see that this sequence goes **dow**n by 3 each time. This means our nth term will start with -3n

**Step 2**: Find the value of b.

'What number would come before the first term?'. In this example, the number would be 10 (the numbers change by 4 each time)

Our Nth term will be -3n + 10.

This could also be written as 10 - 3n